## Re: Litoria: New scenario for the planet's 2 moons

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• ... Yes, thanks. And thanks to Pavel for his help. Unfortunately, the math is a bit beyond me, but I m glad to know there is a solution. As of tomorrow I am
Message 1 of 9 , Dec 31, 2010
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--- In conculture@yahoogroups.com, "tomhchappell" <tomhchappell@...> wrote:
>
> Does any of that help?
>
Yes, thanks. And thanks to Pavel for his help. Unfortunately, the math is a bit beyond me, but I'm glad to know there is a solution. As of tomorrow I am retired and I hope to have more time to devote to my concultures. When I get around to Litoria, I'll make sure I make use of the knowledge you've given me.

Charlie
• Aren t the moons of Mars both about the same size? Both are rather small compared to the planet, though! /bpj
Message 2 of 9 , Jan 2, 2011
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Aren't the moons of Mars both about the same size?
Both are rather small compared to the planet, though!

/bpj

2010-12-30 23:57, tomhchappell skrev:
> Additionally you have to consider the three-body problem.
>
> If the two moons have the same orbit, then there are only five places the smaller one can be, and only two of those are stable orbits.
>
> The mass of the smaller moon has to be negligible compared to the mass of the larger moon; and the mass of the larger moon has to be less than 1/30 the mass of the planet.
>
> The smaller moon can then be:
> 1. on the far side of the larger moon from the planet;
> 2. on the far side of the planet from the larger moon;
> 3. in between the planet and the larger moon, on a straight line;
> 4. in the larger moon's orbit but 60 degrees ahead of it;
> 5. in the larger moon's orbit but 60 degrees behind it.
>
> Only L4 and L5 are stable, and even then only if the mass of the smaller moon is negligible compared to that of the larger moon, and the mass of the larger moon is less than about 1/30 the mass of the planet.
>
> -------------------------------
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> OTOH there are several stable three-body systems in the Solar system in which the ratio of the two smaller bodies' orbital periods is that of two small whole numbers.
>
> Additionally there are such ratios (ratios of two small whole numbers) that seem to be avoided in the Solar System, presumably because they're very unstable.
>
> A ratio of 2:1 is one of the stable ones.
>
> If their orbital periods were in a ratio of 2:1, and the angle between their orbital planes were quite shallow (say, half a degree or so), then they'd appear in the same part of the sky quite often. Many of those times the closer one would partially eclipse the further one; occasionally the closer one would totally eclipse the further one; and occasionally they would both appear in the same part of the sky without any eclipsing, so, if you looked at it from the correct angle, it would look like one was above the other.
>
> Using Kepler's laws that A. da Mek linked to, the further one would need to be 2^(3/2) as far away as the smaller one; 2^(3/2) is about 2.82842712 or so. So, for them to look the same size, the further would also have to have 2^(1.5) times the radius of the nearer. Assuming the same density, the ratio of their masses is the cube of the ratio of their radii; that is, about 2^(4.5) or about 22.627417.
>
> In order to not have to worry about the moons masses, you'd want the gravitational effect the moons have on each other to be less than about 1% of that the planet has on them. The biggest such influence is the larger moon's influence on the smaller moon when they are closest together. Let the mass of the planet be denoted by P, and the mass of the larger moon by M. If M is less than about 1% of P you don't need to worry much. I don't know how big you want the larger moon to be, though. If M is less than 10% of P the calculations might not be too difficult.
>
> You also probably don't have to worry too much about how the moons' gravitational fields effect one anothers' orbits if the larger moon is 5 times as far away as the smaller moon, or further.
>
> Certainly, if the larger one is 11 times further away or further, then when the moons and the planet are all in a straight line with the smaller moon in the middle, the larger moon is 10 times further away from the smaller moon than the planet is; so the larger moon's pull on the smaller moon is less than 1% of the planet's pull on the smaller moon.
>
> And, if we assume that the planet is at least 30 times as massive as the larger moon, and the larger moon is at least 5 times as far away from the planet as the smaller moon, then when they're all lined up with the little one in the middle, the larger moon is at least 4 times further away from the little moon than the planet, so the planet's pull on the little moon is at least 30*4^2 = 480 times the big moon's pull on the little one.
>
> A ratio of orbital periods of 3:1 is one of the stable ones, IIRC, and it corresponds to a ratio of orbital radii of about 5.196152423.
> That, in turn, corresponds to a ratio of masses of about 140.2961154.
>
> Actually, if the planet's mass is at least 30 times the large moon's mass, the ratio of orbital radii needn't be more than 1+sqrt(10/3)= about 2.825741858 for the large moon's pull on the smaller moon to be not over 1% of the planet's pull on the smaller moon. The orbital resonance of 2:1 is just barely big enough for that.
>
> If you wanted a 3:2 resonance the ratio of the orbital radii would need to be about 1.837117307 and so at closest approach the big moon would actually be closer to the small moon than the planet would be. So you'd need the big moon's mass to be less than about 1/70 of the planet's mass.
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> Does any of that help?
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• From: conculture@yahoogroups.com [mailto:conculture@yahoogroups.com] On Behalf Of BPJ --- ... Well, Deimos (1.48x10^15 kg) is a good order of magnitude less
Message 3 of 9 , Jan 2, 2011
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From: conculture@yahoogroups.com [mailto:conculture@yahoogroups.com] On Behalf Of BPJ --->

> Aren't the moons of Mars both about the same size?
> Both are rather small compared to the planet, though!

Well, Deimos (1.48x10^15 kg) is a good order of magnitude less massy than Phobos (1.07 x 10^16 kg), but, yeah, not enough of a difference to count as negligible.

But in their case, it doesn't matter, since they're in anything but the same orbit and so won't (in their case) peturb each other out of it. Deimos is relatively far out and has an almost circular orbit close to the equator, not even all that are off (in astronomical terms), um, areosynchronous. Really very convenient if we ever get around to colonizing the planet, really, since you wouldn't have to do _that_ much to Deimos's orbit to make it a decent anchor for a beanstalk. Phobos, on the other hand, has a rather more eccentric orbit, very, very close in to the planet. (Fun trivia tip: being so close, it orbits so fast it overtakes the planet's spin, rising in the west and setting in the east. _Multiple_ times per day.)

And if we ever build that beanstalk, watching Phobos fly by at close-enough-to-touch distance is going to give some passenger cars a _spectacular_ view. :)

Alistair

2010-12-30 23:57, tomhchappell skrev:
> Additionally you have to consider the three-body problem.
>
> If the two moons have the same orbit, then there are only five places the smaller one can be, and only two of those are stable orbits.

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• ... But, you re not old enough! Only eighty-something, right?
Message 4 of 9 , Jan 3, 2011
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---In conculture@yahoogroups.com, "Charlie" <caeruleancentaur@...> wrote:
>As of tomorrow I am retired
...
> Charlie

But, you're not old enough! Only eighty-something, right?
• I don t know exactly what, in the previous parts of this thread, your question refers to. Maybe it s just something else relevant to the thread? Anyway, see
Message 5 of 9 , Jan 3, 2011
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I don't know exactly what, in the previous parts of this thread, your question refers to.
Maybe it's just something else relevant to the thread?
Anyway, see
<http://en.wikipedia.org/wiki/Deimos_(moon) >
and
<http://en.wikipedia.org/wiki/Phobos_(moon) >

Short answer; No, they're not about the same size. Phobos is bigger, and it's also closer so it looks _much_ bigger.

Phobos is the larger and closer of the two moons of Mars, the other being Deimos. With a mean radius of 11.1 km (6.9 mi), Phobos is 7.24 times as massive as Deimos. Phobos orbits about 9,377 km from the center of Mars, closer to its primary than any other known planetary moon. It orbits so close to the planet that it moves around Mars faster than Mars itself rotates. As a result, from the surface of Mars it appears to rise in the west, move rapidly across the sky (in 4 h 15 min or less) and set in the east.

Deimos is the smaller and outer of Mars's two moons (the other being Phobos). Deimos, like Mars' other moon Phobos, has spectra, albedos and densities similar to those of a C- or D-type asteroid. Deimos' orbit is nearly circular and is close to Mars' equatorial plane. Both Deimos and Phobos have very circular orbits which lie almost exactly in Mars' equatorial plane. As seen from Mars, Deimos would have an angular diameter of no more than 2.5 minutes and would therefore appear almost star-like to the naked eye. At its brightest ("full moon") it would be about as bright as Venus is from Earth; at the first- or third-quarter phase it would be about as bright as Vega. With a small telescope, a Martian observer could see Deimos' phases, which take 1.2648 days (Deimos' synodic period) to run their course. Unlike Phobos, which orbits so fast that it actually rises in the west and sets in the east, Deimos rises in the east and sets in the west. However, the Sun-synodic orbital period of Deimos of about 30.4 hours exceeds the Martian solar day ("sol") of about 24.7 hours by such a small amount that 2.7 days elapse between its rising and setting for an equatorial observer. Because Deimosâ orbit is relatively close to Mars and has only a very small inclination to Marsâ equator, it cannot be seen from Martian latitudes greater than 82.7Â°.

Phobos's characteristics:
Mean radius 11.1 km (0.0021 Earths)
Mass 1.072Ã10^16 kg (1.8 nEarths)
Orbital Semi-major axis 9,377.2 km
Orbital period 0.31891023 d (7 h 39.2 min)

Deimos's characteristics:
Mass 1.48Ã10^15 kg (0.25 nEarths)
Orbital Semi-major axis 23,460 kilometres
Orbital period 1.262 44 d (30.30 hours)

Phobos is not quite twice as wide as Deimos and so since their density is similar it's not quite 8 times as massive. Deimos orbits around 2.5 times as far from Mars as Phobos does. The ratio of their orbital periods is nearly 4.

Phobos would, I guess, appear about 5 times (5 = 2 * 2.5) as big as Deimos when seen from the surface of Mars.

--- In conculture@yahoogroups.com, BPJ <melroch@...> wrote:
>
> Aren't the moons of Mars both about the same size?
> Both are rather small compared to the planet, though!
>
> /bpj
>
• Ah, darn, I got post-ninjaed.
Message 6 of 9 , Jan 3, 2011
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Ah, darn, I got post-ninjaed.

--- In conculture@yahoogroups.com, Alistair J R Young <avatar@...> wrote:
>
> Well, Deimos (1.48x10^15 kg) is a good order of magnitude less massy than Phobos (1.07 x 10^16 kg), but, yeah, not enough of a difference to count as negligible.
>
> But in their case, it doesn't matter, since they're in anything but the same orbit and so won't (in their case) peturb each other out of it.

> Alistair
>
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